Figure 1

Left part: Contour plot of the distribution of a spherical particle which flows around circular post (black circle; for symmetry reasons only the upper half is shown), obtained by numerically integrating the equation of motion (8) to time $t=10\hspace{0.5em}\text{s}$ starting at initial position $\mathrm{x⃗}(0)=(-11.5\hspace{0.5em}\mu \text{m},0\hspace{0.5em}\mu \text{m})$. Further system parameters are: particle radius $a=1\hspace{0.5em}\mu \text{m}$, room temperature $\mathit{kT}=4\hspace{0.5em}\text{fN}\hspace{0.5em}\mu \text{m}$, Stokes friction in water $\eta =18.9\hspace{0.5em}\text{fN}\hspace{0.5em}\text{s}/\mu \text{m}$. The velocity field is given by $\mathrm{v⃗}(\mathrm{x⃗},t)=\mathrm{v⃗}(\mathrm{x⃗})=(\partial \psi /\partial x_2,-\partial \psi /\partial x_1)$ with the stream function $\psi (\mathrm{x⃗})=v_0{x}_2[2\ln (\left|\mathrm{x⃗}\right|/R)-1+R^2/|\mathrm{x⃗}|{}^2]$, representing the (two-dimensional) Stokes flow around a post of radius $R$ located at the origin (we take $R=10\hspace{0.5em}\mu \text{m}$ and $v_0=5\hspace{0.5em}\mu \text{m}/\text{s}$); indices $1$ and $2$ refer to the two spatial directions. The different symbols show results for the different integration schemes with $\mathit{dt}=0.01\hspace{0.5em}\text{s}$ [15]: red stars, proposed algorithm; filled green squares, rejection scheme; open blue diamonds, event-driven scheme (see main text); black solid lines, exact result (obtained from numerically solving the associated Fokker-Planck equation [6]). The gray shaded region is inaccessible to the particle center due to the hard-wall interaction. Right part and inset: Particle diffusion $\langle [({\mathrm{x⃗}}_{0+\mathit{dt}}-{\mathrm{x⃗}}_0)·{\mathrm{e⃗}}_2]{}^2\rangle /(2\mathit{dt})$ tangential to the post wall during the first time step of length $\mathit{dt}=0.01\hspace{0.5em}\text{s}$ as obtained from the different integration algorithms. The exact value is $X=\mathit{kT}/\eta =0.21\hspace{0.5em}\mu {\text{m}}^2/\text{s}$.Reuse & Permissions

Figure 2

Contour plot of the distribution of (a) the relative coordinate $\mathrm{y⃗}-\mathrm{x⃗}$ and (b) the center of friction coordinate $(\eta \mathrm{x⃗}+\gamma \mathrm{y⃗})/(\eta +\gamma )$ for two particles in a shear flow, obtained from numerically integrating the equations of motion (10a) to time $t=1\hspace{0.5em}\text{s}$ (time step $\mathit{dt}=0.005\hspace{0.5em}\text{s}$), with initial positions $\mathrm{x⃗}(0)=(0\hspace{0.5em}\mu \text{m},0\hspace{0.5em}\mu \text{m})$ and $\mathrm{y⃗}(0)=(-2\hspace{0.5em}\mu \text{m},0.75\hspace{0.5em}\mu \text{m})$, particle radii $a=1\hspace{0.5em}\mu \text{m}$, $b=0.5\hspace{0.5em}\mu \text{m}$, room temperature $\mathit{kT}=4\hspace{0.5em}\text{fN}\hspace{0.5em}\mu \text{m}$, Stokes friction in water $\eta =18.9\hspace{0.5em}\text{fN}\hspace{0.5em}\text{s}/\mu \text{m}$, $\gamma =9.45\hspace{0.5em}\text{fN}\hspace{0.5em}\text{s}/\mu \text{m}$, and identical flow fields for both particles, $\mathrm{v⃗}(\mathrm{x⃗},\mathrm{y⃗},t)=\mathrm{v⃗}(\mathrm{x⃗})=\alpha x_2{\mathrm{e⃗}}_1$, $\mathrm{u⃗}(\mathrm{x⃗},\mathrm{y⃗},t)=\mathrm{u⃗}(\mathrm{y⃗})=\alpha y_2{\mathrm{e⃗}}_1$, where $\alpha$ is the shear rate (we take $\alpha =5/\text{s}$) and indices $1$ and $2$ indicate the two spatial directions. The different symbols correspond to the different integration schemes like in Fig. 1 [15]; the gray shaded circle in (a) represents the effective region inaccessible to the particles due to their hard-core repulsion.Reuse & Permissions